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Crossing Number of Simple 3-Plane Drawings

Authors Miriam Goetze , Michael Hoffmann , Ignaz Rutter , Torsten Ueckerdt

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graphs and surfaces
Keywords
  • beyond planar graphs
  • edge density
  • crossing number
  • density formula

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Abstract

We study 3-plane drawings, that is, drawings of graphs in which every edge has at most three crossings. We show how the recently developed Density Formula for topological drawings of graphs [Kaufmann et al., 2024] can be used to count the crossings in terms of the number n of vertices. As a main result, we show that every 3-plane drawing has at most 5.5(n-2) crossings, which is tight. In particular, it follows that every 3-planar graph on n vertices has crossing number at most 5.5n, which improves upon a recent bound [Bekos et al., 2024] of 6.6n. To apply the Density Formula, we carefully analyze the interplay between certain configurations of cells in a 3-plane drawing. As a by-product, we also obtain an alternative proof for the known statement that every 3-planar graph has at most 5.5(n-2) edges.

Cite As Get BibTex

Miriam Goetze, Michael Hoffmann, Ignaz Rutter, and Torsten Ueckerdt. Crossing Number of Simple 3-Plane Drawings. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.GD.2025.15

Author Details

Miriam Goetze
  • Karlsruhe Institute of Technology, Germany
Michael Hoffmann
  • Department of Computer Science, ETH Zürich, Switzerland
Ignaz Rutter
  • University of Passau, Germany
Torsten Ueckerdt
  • Karlsruhe Institute of Technology, Germany

Funding

  • Goetze, Miriam: funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 520723789.
  • Rutter, Ignaz: funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 541433306.
  • Ueckerdt, Torsten: funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 520708409.

Acknowledgements

This research was initiated at the Workshop on Graph and Network Visualization (GNV 2024) in Heiligkreuztal, Germany, June 23-28, 2024.

References

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